## Serenitypaige1129 2 years ago What is a quartic function with only the two real zeroes given? x = –1 and x = –3

1. tcarroll010

You can go with (x + 1)^2 times (x + 3)^2 The reason this works is that it will give you a x^4 term and you still have only the 2 roots. Roots are of the form: (x - root) for you factors.

2. tcarroll010

So, basically you have both roots twice, so that's why each factor has a 2 on it for an exponent.

3. tcarroll010

You could also have put a 3 on one of them and left the other root as a single power (no exponent)

4. tcarroll010

Is this making sense to you?

5. Serenitypaige1129

I am lost completely

6. tcarroll010

np, I'll give you a little more detail: First off, a quartic polynomial function has as its highest exponent, an x^4 term. That means that if you have an equation of the form: y = (x - root1)(x - root2)(x - root3)(x - root4) you will have quartic function. But we can have only 2 different roots, so a good choice is to have root1 twice and root 2 twice: y = [(x - root1)^2][(x - root2)^2]

7. tcarroll010

That's the same as: y = (x - root1)(x - root1)(x - root2)(x - root2) which also has root1 twice and root2 twice. It's just writing out the factors without the exponents. This equation will give you 0 for "y" if you replace x with either root, which is what you want to solve this problem.

8. tcarroll010

And to finish this off, you are given x = -1 and x = -3 as the only 2 roots you can have. And since the factor to give "0" using the root is: (x - root) one factor is [x - (-1)] which is (x + 1) the other is [x - (-3)] which is (x + 3) So, your polynomial is: y = (x + 1)(x + 1)(x + 3)(x + 3)

9. tcarroll010

You have other choices, but the above polynomial works. You could also have: y = (x + 1)(x + 3)(x + 3)(x + 3) or y = (x + 1)(x + 1)(x + 1)(x + 3) Both of those are quartics having only 2 different roots. That should make it all clearer now. Now are you good on this?

10. tcarroll010

Here's a graph of the first solution

11. Serenitypaige1129

sorta so which one would it be? y = x4 – 4x3 – 4x2 – 4x – 3 y = –x4 + 4x3 + 4x2 + 4x + 3 y = x4 + 4x3 + 3x2 + 4x – 4 y = x4 + 4x3 + 4x2 + 4x +3 I understand how you did that i just don't know which one it will be.

12. Serenitypaige1129

@tcarroll010

13. tcarroll010

I didn't know that you had choices. That changes the whole way you approach this problem. You can solve it 2 ways that I know of. The first way rst and the best and fastest is to perfrom synthetic division on each choice until you hit upon the choice that has only those 2 roots.

14. tcarroll010

Number #1 is out because -1 is not a root. So, I eliminated that one so far.

15. tcarroll010

Number 2 is out because it also has +1 as a root.

16. tcarroll010

Are you sure you wrote those choices down correctly?

17. Serenitypaige1129

I actually copied and paste so yeah

18. tcarroll010

And the roots are supposed to be -1 and -3 ?

19. Serenitypaige1129

yeah that is why i couldn't figure it out. That is what confused me

20. tcarroll010

It looks like like the last one works. I tried it again.

21. Serenitypaige1129

okay thanks again

22. tcarroll010

you're welcome.